This document will allow university Math students to practice integration by trigonometric substitution. All the questions are similar in style and full worked solutions have been provided. This will allow you to understand how a certain question in this document is answered and then you will be ab...
Calculus 2 - Integration by Trigonometric Substitution
Case 1:
∫ 𝑎² − 𝑥² 𝑑𝑥
Use substitution 𝑥 = 𝑎𝑠𝑖𝑛𝜃
𝑑𝑥 = 𝑎𝑐𝑜𝑠𝜃𝑑𝜃
Case 2:
∫ 𝑥² − 𝑎² 𝑑𝑥
Use Substitution 𝑥 = 𝑎𝑠𝑒𝑐𝜃
𝑑𝑥 = 𝑎𝑠𝑒𝑐𝜃𝑡𝑎𝑛𝜃𝑑𝜃
Case 3:
∫ 𝑥² + 𝑎² 𝑑𝑥 OR ∫ 𝑎² + 𝑥² 𝑑𝑥
Use Substitution 𝑥 = 𝑎𝑡𝑎𝑛𝜃
𝑑𝑥 = 𝑎𝑠𝑒𝑐²𝜃𝑑𝜃
, 1) ∫ 49 − 𝑥² 𝑑𝑥
Case 1:
𝑎² = 49
𝑎 = 49
𝑎 = 7
𝑥 = 7𝑠𝑖𝑛𝜃
𝑑𝑥 = 7𝑐𝑜𝑠𝜃𝑑𝜃
Substitute 𝑥 = 7𝑠𝑖𝑛𝜃 and 𝑑𝑥 = 7𝑐𝑜𝑠𝜃𝑑𝜃 into the original integral
∫ 49 − (7𝑠𝑖𝑛𝜃)² 7𝑐𝑜𝑠𝜃 𝑑𝜃
This integral is now in trigonometric terms
∫ 49 − 49𝑠𝑖𝑛²𝜃 7𝑐𝑜𝑠𝜃 𝑑𝜃
∫ 49(1 − 𝑠𝑖𝑛²𝜃) 7𝑐𝑜𝑠𝜃 𝑑𝜃
∫ 49𝑐𝑜𝑠²𝜃 7𝑐𝑜𝑠𝜃 𝑑𝜃
∫7𝑐𝑜𝑠𝜃 7𝑐𝑜𝑠𝜃 𝑑𝜃
∫49𝑐𝑜𝑠²𝜃 𝑑𝜃
49∫𝑐𝑜𝑠²𝜃 𝑑𝜃
Power reduction formula for cos²𝜃:
1 1
𝑐𝑜𝑠²𝜃 = 2
+ 2
𝑐𝑜𝑠2𝜃
1 1
49∫ 2 + 2
𝑐𝑜𝑠2𝜃 𝑑𝜃
1 1
49 ( 2 𝜃 + 4
𝑠𝑖𝑛2𝜃)
49 49
2
𝜃 + 4
𝑠𝑖𝑛2𝜃
,The expression in the green box is the answer however, we need to convert this
answer to an expression in terms of 𝑥 since the original question was in terms of 𝑥.
We will add the + C at the very end in the final answer to avoid any confusion.
Sin Double Angle Formula: 𝑠𝑖𝑛2𝜃 = 2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃
49 49
2
𝜃 + 4
(2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃)
49 49
2
𝜃 + 2
𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃
𝑥 = 7𝑠𝑖𝑛𝜃 → This is the first substitution we used at the beginning of the question.
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller alikhalid2. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £9.16. You're not tied to anything after your purchase.